Tilt Series CTF Estimation

Contrast transfer function estimation in tilt series

From Real-time cryo-electron microscopy data preprocessing with Warp

The single micrograph CTF estimation procedure with planar sample geometry described in the previous section can be used for tilted 2D data collection. However, full tilt series pose additional challenges for CTF fitting. Mechanical stage instabilities and imperfect eucentric height setting necessitate additional exposures for tracking and focusing to correct the stage position between individual tilt images. Thus, the defocus cannot be assumed to stay constant, or change smoothly over the course of a tilt series. Each tilt image requires its own defocus value, which can be challenging due to the small amount of signal available. Even at 120 e- per Ų for an entire series of 60 images, each tilt only has 2 e- per Ų to perform the same estimation as for a 40 e- per Ų 2D image, while striving to achieve comparable accuracy.

CTF estimation in tilt series has traditionally received less attention than its equivalent in 2D data, with the most recent publication predating the introduction of direct electron detectors and phase plates. As the resolution obtainable through sub-tomogram averaging has come close to parity with 2D data since then, simplifying assumptions such as the neglect of astigmatism or the assumed flatness of the sample can limit the resolution. Combined with the lack of integration of dedicated tilt series CTF estimation tools into common sub-tomogram averaging pipelines, this has created a situation in which state-of-the-art studies employ tools designed for 2D data, such as CTFFIND.

To improve the fit accuracy, the individual tilt image fits must be subjected to a common set of constraints. As the imaged sample content does not change significantly throughout the tilt series, 1D background and envelope can be derived from the average 1D spectrum of all tilt images. The relative tilt angles and the tilt axis orientation are known to a higher precision than could be derived from fitting a planar geometry de novo, and are kept constant throughout the optimization, as suggested previously. However, the absolute inclination of the sample plane is unknown. This is especially critical in some of the typical applications of tomography, like the imaging of lamellae prepared through focused ion beam (FIB) milling because a lamella can be tilted by over 20° relative to the grid. This additional inclination remains constant throughout the tilt series, and is made a single optimizable parameter for all tilt images. Astigmatism and, optionally, phase shift can be kept constant throughout 2D image exposures, but can benefit from a temporally resolved model in a tilt series where the overall exposure is fractionated over a much longer time, for example 20–30 min. Warp uses these three control points in the temporal dimension to model these parameters.

With these considerations, the full estimation process is as follows. 2D patches are extracted from all aligned tilt movie averages, as described in the micrograph CTF fitting procedure, and treated in parallel in all subsequent calculations. To provide a better initialization for the per-tilt defocus searches, an estimate for the average defocus of the entire series is obtained by prepping 1D spectra from all patches, and comparing them to simulated CTF curves for the defocus values at the respective positions and tilts, taking into account the fixed relative tilt information and the currently tested average defocus (and phase shift, optionally). This search is performed exhaustively over a range of values specified by the user. The result is used as the starting point of a more complex optimization. Defocus values for all individual tilts, three astigmatism magnitude–angle pairs, three optional phase shift values, and the two global inclination angles (that is the plane normal) are optimized using the L-BFGS algorithm with the derivatives obtained numerically as described in the micrograph CTF fitting section. Upon convergence, the 1D spectra of all patches are rescaled to a common defocus value. This is especially useful for validation in tilt series since the individual images will have very noisy spectra. If the useful resolution range does not extend sufficiently beyond the fitting range, the latter is automatically decreased and the optimization repeated.

In our experience, the direction of the tilt axis is often miscalibrated. Correct handedness in structures obtained from sub-tomogram averaging does not guarantee the tilt angle sign is not flipped. In Warp, a positive rotation around the positive Y image axis is defined to result in an increased underfocus at positions to the positive X side of the tilt axis, that is those parts of the sample comically closer to the electron beam source. The CTF fitting procedure in Warp can detect such mistakes by optionally repeating the optimization with the tilt angles flipped, and notifying the user if the ' wrong' set of angles provides a better fit. Such a test can be used to re-calibrate the acquisition software for future data collection.